Setting "a"="bi" where "i" is the imaginary unit, and applying Euler's identity, one sees that the Weierstrasstransform of the function cos("bx") is "e" cos("bx") and the Weierstrasstransform of the function sin("bx") is "e" sin("bx").

The above formal derivation glosses over details of convergence, and the formula "W" = "e" is thus not universally valid; there are several functions "f" which have a well-defined Weierstrasstransform, but for which "e""f"("x") cannot be meaningfully defined.

The Weierstrasstransform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, then the temperature distribution of the rod "t" = 1 time units later will be given by the function "F". By using values of "t" different from 1, we can define the generalized Weierstrasstransform of .